Numerical studies of rotating bose einstein condensates via rotating lagrangian coordinates

Numerical Studies of Rotating
Bose-Einstein Condensates via
Rotating Lagrangian Coordinates

CAO XIAOMENG
(B.Sc.(Hons.), NUS,
Diplˆome de l’Ecole Polytechnique)

A THESIS SUBMITTED FOR THE DEGREE OF
MASTER OF SCIENCE
DEPARTMENT OF MATHEMATICS
NATIONAL UNIVERSITY OF SINGAPORE
2012


i


Acknowledgements

I would like to extend my heartfelt gratitude to my supervisor: Prof
Bao Weizhu, for his guidance and advice throughout the research
process. He has provided me great ideas and strong encouragement.

In addition, I’d like to express special thanks to Mr. Tang Qinglin
for all his help and guidance.

Moreover, I’d like to thank the professors in National University
of Singapore and Ecole Polytechnique in France, for their guidance
through out my two years studies in Singapore and two years studies
in France under the French-Grandes Ecoles Double Degree Program

(FDDP). I’d like to thank Assoc. Prof. Wong Yan Loi for his coordination and help in FDDP during the years.

And finally, I wish to thank my parents for their understanding,
unconditional support and sacrifice over the years. I would never
arrive at this stage without them.


Contents
Contents

iii

List of Tables

vii

List of Figures

ix

1 Introduction

1

1.1

Bose-Einstein condensates . . . . . . . . . . . . . . . . . . . . . .

1

1.2

The Gross-Pitaevskii equation . . . . . . . . . . . . . . . . . . . .

3

1.3

Existing numerical methods . . . . . . . . . . . . . . . . . . . . .

4

1.4

Purpose of the study and structure of the thesis . . . . . . . . . .

5

2 Methods and analysis for rotating BEC

7

2.1

Dynamical laws in the Cartesian coordinate . . . . . . . . . . . .

7

2.2

GPE under a rotating Lagrangian coordinate . . . . . . . . . . .

10

2.3

Dynamical laws in the Lagrangian coordinate . . . . . . . . . . .

11

2.4

Numerical methods . . . . . . . . . . . . . . . . . . . . . . . . . .

21

2.4.1

Time splitting method . . . . . . . . . . . . . . . . . . . .

21

2.4.2

Discretization in 2D . . . . . . . . . . . . . . . . . . . . .

23

2.4.3

Discretization in 3D . . . . . . . . . . . . . . . . . . . . .

24

Numerical results . . . . . . . . . . . . . . . . . . . . . . . . . . .

24

2.5.1

Accuracy test . . . . . . . . . . . . . . . . . . . . . . . . .

25

2.5.2

Dynamical results in 2D . . . . . . . . . . . . . . . . . . .

26

2.5.3

Quantized vortex interaction in 2D . . . . . . . . . . . . .

33

2.5

iii

CONTENTS

3 Extention to rotating two-component BEC

45

3.1

Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

45

3.2

Coupled Gross-Pitaevskii equations . . . . . . . . . . . . . . . . .

46

3.3

Dynamical laws in the Cartesian coordinate . . . . . . . . . . . .

47

3.4

The Lagragian transformation . . . . . . . . . . . . . . . . . . . .

50

3.5

Dynamical laws in the Lagrangian coordinate . . . . . . . . . . .

51

3.6

Numerical methods . . . . . . . . . . . . . . . . . . . . . . . . . .

62

3.7

Numerical results . . . . . . . . . . . . . . . . . . . . . . . . . . .

64

3.7.1

Dynamics of energy and density

. . . . . . . . . . . . . .

64

3.7.2

Conservation of angular momentum expectation . . . . .

65

3.7.3

Center of mass . . . . . . . . . . . . . . . . . . . . . . . .

65

3.7.4

Condensate width . . . . . . . . . . . . . . . . . . . . . .

67

3.7.5

Dynamics of vortex lattices . . . . . . . . . . . . . . . . .

68

4 Conclusion and future studies

77

Bibliography

79

iv

Summary

Since the realization of Bose-Einstein Condensation (BEC) in dilute bosonic atomic gases [54, 5, 16], significant experimental and
theoretical advances have been developed in the field of research
[56, 64, 4, 3, 13, 35, 33] which permits an intriguing glimpse into
the macroscopic quantum world. Quantized vortices in rotating BEC
have been observed by several groups experimentally, e.g. the JILA
group [56], the ENS group [54], and the MIT group [64]. There are
various methods to generate quantized vortices, including imposing
a rotating laser beam with angular velocity on the magnetic trap [21]
and adding a narrow, moving Gaussian potential to the stationary
magnetic trap [45]. These observations have spurred great excitement in studying superfluidity.
In this thesis, the dynamics of rotating BEC is studied analytically
and numerically based on introducing a rotating Lagrangian coordinate. Based on the mean field theory, the rotating one-component
BEC is described by a single Gross-Pitaevskii equation (GPE) in
a rotating frame. By introducing a rotating Lagrangian coordinate
transform, the angular momentum term has been removed from the
original GPE and is replaced by a time-dependent potential. We
find the formulation for energy and proved its conservation. And
we study the conservation and dynamical laws of angular momentum expectation and condensate width. We investigate the center
of mass with initial ground state with a shift. A numerical method,
which is explicit, stable, spectral accurate is presented. Extensive numerical results are presented to demonstrate the dynamical results.


Finally, these numerical results are extended to two-component rotating BEC.
The first chapter of this thesis will focus on the background of BEC
and existing numerical methods. The work in this thesis will be introduced as well.
Chapter 2 will focus on the single-component BEC. We apply the

coordinate transformation methodology. The dynamical laws of the
rotating BEC under the new coordinate system will be discussed
and presented in details. We approximate the rotating BEC using
time splitting method for temporal direction and spectral discretization method for spatial direction. Numerical results will also be presented.
Our investigation is extended to two-component rotating BEC in
Chapter 3. We apply a similar approach to the coupled GPE where
the dynamics is studied both analytically and numerically.


List of Tables
2.1

Spatial error analysis: Error ||φe (t) − φh,k (t)||l2 at t = 2.0 with
k = 1E − 4. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

2.2

25

Temporal error analysis: Error ||φe (t) − φh,k (t)||l2 at t = 2.0 with
h = 1/32. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

vii

26


LIST OF TABLES

viii

List of Figures
1.1

Velocity-distribution data of a gas of Rubidium (Rb) atoms, confirming the discovery of a new phase of matter, the Bose-Einstein
condensate. Left: just before the appearance of a Bose-Einstein
condensate. Center: just after the appearance of the condensate.
Right: after further evaporation, leaving a sample of nearly pure
condensate. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

2

2.1

Dynamics of mass and energies under Ω = 0, γx = γy = 1. . . . .

28

2.2

Dynamics of mass and energies under Ω = 0, γx = 1, γy = 8. . . .

28

2.3

Dynamics of mass and energies under Ω = 1, γx = γy = 1. . . . .

29

2.4

Dynamics of mass and energies under Ω = 4, γx = 1, γy = 2. . . .

29

2.5

Dynamics of condensate width and angular momentum under Ω =
1, γx = γy = 1, x0 = 1, y0 = 1. . . . . . . . . . . . . . . . . . . . .

2.6

Dynamics of condensate width and angular momentum under Ω =
1, γx = 1, γy = 2, x0 = 0, y0 = 1. . . . . . . . . . . . . . . . . . . .

2.7

31

Dynamics of condensate width and angular momentum under Ω =
0, γx = 1, γy = 2, x0 = 1, y0 = 1. . . . . . . . . . . . . . . . . . . .

2.9

30

Dynamics of condensate width and angular momentum under Ω =
0, γx = 1, γy = 2, x0 = 0, y0 = 1. . . . . . . . . . . . . . . . . . . .

2.8

30

31

Trajectory of center of mass under original and transformed frame
when γx = γy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

35

2.10 Trajectory of center of mass under original and transformed frame
when Ω = 0, γx = 1, γy = 8. . . . . . . . . . . . . . . . . . . . . .

ix

35


LIST OF FIGURES

2.11 Trajectory of center of mass under original and transformed frame
when Ω = 0, γx = 1, γy = 2π. . . . . . . . . . . . . . . . . . . . .

35

2.12 Trajectory of center of mass under original frame when Ω =
1/5, γx = γy = 1. . . . . . . . . . . . . . . . . . . . . . . . . . . .

36

2.13 Trajectory of center of mass under original frame when Ω =
4/5, γx = γy = 1. . . . . . . . . . . . . . . . . . . . . . . . . . . .

36

2.14 Trajectory of center of mass under original frame when Ω =
1, γx = γy = 1. . . . . . . . . . . . . . . . . . . . . . . . . . . . .

36

2.15 Trajectory of center of mass under original frame when Ω =
3/2, γx = γy = 1. . . . . . . . . . . . . . . . . . . . . . . . . . . .

37

2.16 Trajectory of center of mass under original frame when Ω =
6, γx = γy = 1. . . . . . . . . . . . . . . . . . . . . . . . . . . . .

37

2.17 Trajectory of center of mass under original frame when Ω =
π, γx = γy = 1. . . . . . . . . . . . . . . . . . . . . . . . . . . . .

37

2.18 Trajectory of center of mass under transformed frame when Ω =
1, γx = 1, γy = 2, (x0 , y0 ) = (1, 1). . . . . . . . . . . . . . . . . . .

38

2.19 Trajectory of center of mass under original frame when Ω =
1, γx = 1, γy = 2, (x0 , y0 ) = (1, 1). . . . . . . . . . . . . . . . . . .

38

2.20 Trajectory of center of mass under transformed frame when Ω =
1/2, γx = 1, γy = 2. . . . . . . . . . . . . . . . . . . . . . . . . . .

39

2.21 Trajectory of center of mass under original frame when Ω =
1/2, γx = 1, γy = 2. . . . . . . . . . . . . . . . . . . . . . . . . . .

39

2.22 Trajectory of center of mass under transformed frame when Ω =
4, γx = 1, γy = 2. . . . . . . . . . . . . . . . . . . . . . . . . . . .

40

2.23 Trajectory of center of mass under original frame when Ω =
4, γx = 1, γy = 2. . . . . . . . . . . . . . . . . . . . . . . . . . . .

40

2.24 Trajectory of center of mass under transformed frame when Ω =
1/2, γx = 1, γy = π. . . . . . . . . . . . . . . . . . . . . . . . . . .

40

2.25 Trajectory of center of mass under original frame when Ω =
1/2, γx = 1, γy = π. . . . . . . . . . . . . . . . . . . . . . . . . . .

x

41


LIST OF FIGURES

2.26 Trajectory of center of mass under transformed frame when Ω =
4, γx = 1, γy = π. . . . . . . . . . . . . . . . . . . . . . . . . . . .

41

2.27 Trajectory of center of mass under original frame when Ω =
4, γx = 1, γy = π. . . . . . . . . . . . . . . . . . . . . . . . . . . .

41

2.28 Case I density contour plot, x01 = (0.5, 0), x 02 = (−0.5, 0), (m1 , m2 ) =
(1, 1). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

42

2.29 Case II density contour plot, x01 = (0.5, 0), x 02 = (0, 0), (m1 , m2 ) =
(1, 1). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

42

2.30 Case III density contour plot, x01 = (0.5, 0), x 02 = (−0.5, 0), (m1 , m2 ) =
(1, −1). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

43

2.31 Case IV density contour plot, x01 = (0.5, 0), x 02 = (0, 0), (m1 , m2 ) =
(1, −1). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3.1

Dynamics of total density and density of each component for case
i. (left) and case ii. (right). . . . . . . . . . . . . . . . . . . . . .

3.2

1

(t) (‘-*’) and

˜z
L

2

(t) (‘-o’) when λ = 0 and γx,j = γy,j

for j = 1, 2. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

1

(t) (‘-*’) and

˜z
L

2

(t) (‘-o’) when λ = 0 and γx,j = γy,j

for j = 1, 2. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

66

Dynamics of angular momentum expectation Lz (t) (solid line),
˜z
L

1

(t) (‘-*’) and

˜z
L

2

(t) (‘-o’) when λ = 0 and γx,j = γy,j

for j = 1, 2. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

3.5

66

Dynamics of angular momentum expectation Lz (t) (solid line),
˜z
L

3.4

64

Dynamics of angular momentum expectation Lz (t) (solid line),
˜z
L

3.3

43

66

Time evolution of density surfaces for component one (left) and
component two (right) at different times for case I. From top to
bottom: t = 0, 5, 10, 15.

3.6

. . . . . . . . . . . . . . . . . . . . . . .

69

Time evolution of density surfaces for component one (left) and
component two (right) at different times for case II. From top to
bottom: t = 0, 2.5, 5, 7.5. . . . . . . . . . . . . . . . . . . . . . . .

xi

70


LIST OF FIGURES

3.7

Time evolution of density surfaces for component one (left) and
component two (right) at different times for case III. From top to
bottom: t = 0, 2.5, 5, 7.5. . . . . . . . . . . . . . . . . . . . . . . .

3.8

71

Dynamics of center of mass. Left: trajectory of total center of
mass. Right: the time evolution of center of mass of component
one (top), time evolution of center of mass of component two
(bottom). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

3.9

72

Dynamics of center of mass. Left: trajectory of total center of
mass. Right: the time evolution of center of mass of component
one (top), time evolution of center of mass of component two
(bottom). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

72

3.10 Dynamics of condensate widths σx (t), σy (t) and σr (t) when λ = 0
and V1 (x) = V2 (x). . . . . . . . . . . . . . . . . . . . . . . . . . .

73

3.11 Dynamics of condensate widths σx (t), σy (t) and σr (t) when λ = 0
and V1 (x) = V2 (x). . . . . . . . . . . . . . . . . . . . . . . . . . .

73

3.12 Dynamics of condensate widths σx (t), σy (t) and σr (t) when λ = 0
and V1 (x) = V2 (x). . . . . . . . . . . . . . . . . . . . . . . . . . .

73

3.13 Dynamics of vortex lattices when N = 4 for component one (left)
and component two (right); From top to bottom, t = 0, 0.7, π/2, 2.3, π. 74
3.14 Dynamics of vortex lattices when N = 9 for component one (left)
and component two (right); From top to bottom, t = 0, 0.7, π/2, 2.3, π. 75

xii

Chapter 1

Introduction
1.1

Bose-Einstein condensates

A Bose-Einstein condensate (BEC) is a state of matter of a dilute gas of weakly
interacting bosons which is cooled to temperatures near absolute zero and confined in an external potential. Under these conditions, quantum effects become
apparent on a macroscopic scale, as a large fraction of bosons will spontaneously
occupy the lowest quantum state of the external potential [41].
The idea of BEC was first predicted by Albert Einstein in 1924. He has predicted the existence of a singular quantum state produced by the slowing of
atoms using cooling apparatus [31]. He reviewed and generalized the work of
Satyendra Nath Bose [14] on the statistical mechanics of photons. The result
of the combined efforts of Bose and Einstein forms the concept of a Bose gas,
governed by Bose-Einstein statistics, which describes the statistical distribution
of identical particles with integer spin, known as bosons. In 1938, Fritz London
proposed BEC as a mechanism for superfluidity in liquid helium and superconductivity [15, 53]. Superfluid helium has many exceptional properties, including
zero viscosity and the existence of quantized vortices. It was later discovered
that these properties also appear in the gaseous BEC, after the first experimental realization of BEC, by Eric Cornell, Carl Wieman and co-workers at JILA
on June 5, 1995 in vapours of

87 Rb

(cf. Fig. 1.1) [5]. About four months later,

an independent effort led by Wolfgang Ketterle at MIT created a condensate

1


1. INTRODUCTION

made of

23 Na

[29]. The condensate had about a hundred more atoms, allowing

him to obtain several important experimental results, such as the observation
of quantum mechanical interference between two different condensates. Cornell,
Wieman and Ketterle won the 2011 Nobel Prize in Physics for their achievements. One month after the JILA work, a group led by Randall Hulet at Rice
University announced the create of a condensate of 7 Li atoms [16]. Later, it
was achieved in many other alkali gases, including 85 Rb [27],41 K [57],

133 Cs

[73],

spin-polarized hydrogen [36] and metastable triplet 4 He [65, 67]. These systems
have become a subject of explosion of research.
The most striking feature of BEC is that due to the condensation of a large

Figure 1.1: Velocity-distribution data of a gas of Rubidium (Rb) atoms, confirming the discovery of a new phase of matter, the Bose-Einstein condensate. Left:
just before the appearance of a Bose-Einstein condensate. Center: just after
the appearance of the condensate. Right: after further evaporation, leaving a
sample of nearly pure condensate.

fraction of identical atoms into the same quantum state, the wave-like behaviour
is exhibited on a macroscopic scale, which is distinguishable to the behaviours of
particles following classical Newton’s second law. Another intriguing property
is the unrestricted flow of particles in the sample, such as the flow of currents
without observable viscosity and the flow of electric currents without observable

2


resistance [75]. These properties can be explained by the macroscopic occupation
of a quantized mode which provides a stabilized mechanism. Many experiments
have been carried out to study the superfluid properties of BEC, which has a particularly interesting signature of supporting quantized vortex states [35, 48, 72].
In 1999, a vortex was first created experimentally at JILA using 87 Rb containing
two different hyperfine components [56]. Soon after, the ENS group has created
vortex in elongated rotating cigar-shaped one component condensate, with small
vortex arrays of up to 11 vortices being observed [24, 26, 54, 55]. Recently, MIT
group has created larger rotating condensates with up to 130 vortices being observed [1]. More recently Leanhardt et al have created a coreless vortex in a
spinor F = 1 BEC using “topological phase imprinting” [51]. It is hence of great
importance to study the quantized vortex state to better understand the above
observations as well as superfluidity [35, 48, 68, 17]. Quantized vortex states
can be detected in the experiments of rotating single-component BEC, rotating
two-component and spin F = 1 BEC. Mean-field theory is widely applied to approximate the BEC. The main idea of the theory is to replace interactions of all
particles in the system to any one body with an average or effective interaction,
sometimes called a molecular field [23]. The multi-body problem can be reduced
into a one-body problem. In this case, the interactions between particles in a
dilute atomic gas are very weak and the system can be regarded as being dominated by the wave-like condensate. One can hence apply the main-field theory
and sum the interaction of all of the particles to get an effective one-body problem, which can be approximated using Gross-Pitaevskii equation (GPE) [28, 30].

1.2

The Gross-Pitaevskii equation

The Gross-Pitaevskii equation was first derived in the early 1960s and named
after Eugene P. Gross [42] and Lev Petrovich Pitaevskii [62]. According to theory,
the rotating one-component condensate can be described by a single GPE in a
rotating frame [4, 12, 21, 35, 39, 37]. At temperature T which is much smaller

3


1. INTRODUCTION

than the critical temperature Tc , a BEC could be described by the macroscopic
wave function ψ := ψ(˜
x, t), whose evolution is governed by a self-consistent,
mean field nonlinear Schr¨
odinger equation (NLSE) in a rotational frame, also
known as the GPE with the angular momentum rotation term [21, 8, 34]:



i∂t ψ = − 12 ∇2 ψ + Vhos (˜
x)ψ − ΩLz˜ψ + β|ψ|2 ψ,








 ψ(˜
x, 0) = ψ 0 (˜
x),

(1.1)

˜ ∈ Rd , d = 2, 3.
x

˜ = (˜
˜ = (˜
Here, x
x, y˜)T 2D and x
x, y˜, z˜)T in 3D, is the Cartesian coordinate
vector, and t is time, ψ = ψ(˜
x, t) is the dimensionless wave function, Vhos (˜
x)
is the dimensionless homogeneous external trapping potential, which is often
˜2 + γy2 y˜2 ) in 2D, and resp., Vhos (˜
x) =
harmonic and thus can be written as 12 (γx2 x
1 2 2
˜
2 (γx x

x∂y −
+ γy2 y˜2 + γz2 z˜2 ) in 3D, with γx > 0, γy > 0 and γz > 0. Lz˜ = −i(˜

y˜∂x ) = iJz˜ is the z−component of the angular momentum. Ω is the dimensionless
angular momentum rotating speed. β is a constant characterizing the particle

interactions. When Ω = 0, (1.1) is the expression for single-component nonrotating BEC. The rotating two-component condensates are governed by the
coupled GPEs [38, 40, 48, 58, 69]. The coupled GPEs for two-component BEC
will be discussed in Chapter 3.

1.3

Existing numerical methods

To study the dynamics of BEC, it is essential to have an efficient and accurate numerical method to analyse the time-dependent GPE. In literature, many
numerical methods have been proposed to study the dynamics of non-rotating
single component BEC, which can be grouped into two types. One is finite difference method, such as explicit finite difference method [22], Crank-Nicolson
finite difference method [66] and alternating direction method [71].

Gener-

ally, the accuracy can be second or fourth order in space. The other method
is pseudo-spectral method, for example, Bao [13] has proposed a fourth-order
time-splitting Fourier pseudo-spectral method (TSSP) and a fourth-order time-

4


splitting Laguerre-Hermite pseudo-spectral method (TSLH) [11], Adhikari et
al. have proposed Runge-Kutta pseudospectral method [2, 59]. Researches have
demonstrated that pseudospectral method is more accurate and stable than finite
difference method. However, for a rotating BEC, due to the appearance of the
angular rotating term, the above methods can no longer be applied directly. Limited numerical methods have been proposed to study the dynamics of rotating
BEC, but they are usually low-order finite difference methods [4, 20, 19, 75, 18].
Some better performed methods were designed, for example, Bao et al. [8] has
proposed a numerical method by decoupling the nonlinearity in the GPE and

adopting the polar coordinates or cylindrical coordinates to make the angular
rotating term constant. It is of spectral accuracy in transverse direction but of
second or fourth-order accuracy in radial direction. Another leap-frog spectral
method is proposed, which is of spectral accuracy in space and second-order
accuracy in time [76]. However, it has a stability time constraint for time step
[76].
For coupled-GPEs, there have also been quite a few existing numerical methods,
such as finite difference method and pseudospectral method [6, 38, 25]. But
for rotating coupled-GPEs, due to the rotational term, difficulties have been
introduced as the case for single-component BEC.

1.4

Purpose of the study and structure of the thesis

Hence, it is of a strong interest to develop an accurate, stable and efficient numerical method. In this paper, we have proposed such a numerical method and
studied the dynamics of the rotating BEC by using it. The key feature of the
method is: By taking an orthogonal time-dependent Lagrangian transformation,
the rotational term in GPE can be eliminated under the new rotating Lagrangian
coordinate. We can therefore apply previous numerical methods proposed for
non-rotating BEC on the transformed GPE. In this paper, we have studied the
rotating single component BEC and rotating two-component BEC. We have applied a second-order time splitting method and spectral method in space, which

5


1. INTRODUCTION

is very efficient and accurate. New forms of energy, angular momentum as well
as center of mass for the transformed rotating BEC are defined and presented

both analytically and numerically.
The paper is organized as follows: In Chapter 2, we analyse the rotating single
component BEC. And in Chapter 3, we extend the numerical methods to twocomponent BEC. The two chapters follow the same structure. We first begin by
presenting the dynamical laws of the BEC. We proceed to apply the orthogonal
time dependent matrix transformation method and study the dynamics of the rotating BEC under the new Lagrangian coordinates, we redefined and studied the
conservation of density and energy, as well as angular momentum conservation
under certain conditions. Dynamical laws for condensate width and analytical
solutions for center of mass are presented as well. We follow by presenting an
efficient and accurate numerical method for the simulation of transformed rotating BEC. Numerical results after applying the numerical method are discussed
in section 4, which include accuracy test, dynamical results and quantized vortex
interaction. Finally, some conclusion and further study directions are drawn.

6


Chapter 2

Methods and analysis for
rotating BEC
In this chapter, we study the dynamics of rotating single-component BoseEinstein condensation (BEC) based on the Gross-Pitaevskii equation (GPE)
with an angular momentum rotational term. We first begin by reviewing the
dynamical laws of the BEC according to previous researches. We follow by taking a rotating Lagrangian coordinate which as a result, removes the angular
rotational term in the GPE. We proceed to redefine and study the dynamical
laws of the BEC under the new coordinate system, such as density, energy, angular momentum expectation, condensate width and center of mass. Finally, an
accurate and efficient numerical method is demonstrated and various numerical
results have been presented.

2.1

Dynamical laws in the Cartesian coordinate

There have been many researches done to study the dynamics of rotating BEC
[76, 8], we present a brief review of the dynamical laws of rotating BEC in the
Cartesian coordinate.
I). Energy and density.
There are two important invariants: density and energy and they are defined as

7


2. METHODS AND ANALYSIS FOR ROTATING BEC

follows:
N (ψ) =
Rd

|ψ(˜
x, t)|2 d˜
x,

t ≥ 0,

1
β
|∇ψ|2 + Vhos (˜
x, t)|ψ|2 + |ψ|4 − Ωψ ∗ Lz˜ψ d˜
x,
2
2

E(ψ) =
Rd

(2.1)
(2.2)

where ψ ∗ denotes the complex conjugate of ψ.
II). Angular momentum expectation.
Angular momentum expectation is defined as:

Lz˜ (t) :=

Rd

ψ ∗ Lz˜ψ d˜
x,

t ≥ 0,

d = 2, 3.

(2.3)

Theorem 2.1.1.
γx2 − γy2
d Lz˜ (t)
=
η˜(t),
dt
2

(2.4)

with
η˜(t) :=
R2

x
˜y˜|ψ(˜
x, t)|2 d˜
x.

(2.5)

Thus, we have the conservation of angular momentum expectation and energy
for the non-rotating part, at least in the following cases:
(i): For any given initial data, if we have γx = γy , i.e. the trapping potential is
radially symmetric.
(ii): For any given γx , γy , if we have Ω = 0 and the initial data ψ 0 is even in
either x or y.

III). Condensate width
We define the condensate width as follows along the α-axis (α = x, y, z for 3D),
to quantify the dynamics of the problem (2.13):

δα (t) =

δα (t), with δα = α2 (t) =

Rd

α2 |ψ|2 d˜
x, α = x
˜, y˜, z˜.

(2.6)

We have the following dynamical law for the condensate width:

Theorem 2.1.2. i) Generally, for d=2,3, with any potential and initial data,

8


the condensate width satisfies:
d2 δα (t)
d2 t

=
Rd

(∂y˜α − ∂x˜ α)(4iΩψ ∗ (˜
x∂y˜ + y˜∂x˜ )ψ + 2Ω2 (˜
x2 − y˜2 )|ψ|2 )

+2|∂α ψ|2 + β|ψ|4 − 2α|ψ|2 ∂α Vhos d˜
x,
with
δα (0) =: δα(0) =
δ˙α (0) =: δα(1) = 2

Rd

Rd

α2 |ψ 0 |2 d˜
x,

α Im((ψ 0 )∗ ∂α ψ 0 ) − Ω|ψ 0 |2 (˜
x∂y˜ − y˜∂x˜ )α d˜
x.

ii) In 2D with a radially symmetric trap, i.e. d = 2, γx = γy := γr , we have

(0)
Lz˜ (0)


δr (t) = E(ψ0 )+Ω
[1 − cos(2γr t)] + δr cos(2γr t) +

2
γ

r

(0)
x2 + y˜2 )|ψ 0 |2 d˜
x,
δr (0) =: δr = δx˜ (0) + δy˜(0) = Rd(˜





 δ˙r (0) =: δr(1) = δ˙x˜ (0) + δ˙y˜(0).

(1)

δr
2γr

sin(2γr t),
(2.7)

Moreover, if ψ0 (˜
x) = f (r)eimθ , with m ∈ Z and f (0) = 0 when m = 0, we have,
for any t ≥ 0,
(1)

δ
E(ψ0 ) + mΩ
1
(0)
[1 − cos(2γx t)]+δx˜ cos(2γx t)+ x˜ sin(2γx t).
δx˜ (t) = δy˜(t) = δr (t) =
2
2
2γx
2γx
(2.8)

iii) For all other cases, we have , for t ≥ 0,
(1)

E(ψ0 ) + Ω Lz (0)
δα
δα (t) =
[1 − cos(2γα t)]+δα(0) cos(2γα t)+
sin(2γα t)+fα(t),
2
γα
2γα
(2.9)
where f˜α (t) is the solution of the following equations:
d2 f˜α (t)
+ 4γα2 f˜α (t) = F˜α (t),
d2 t

df˜α (0)
= 0,
f˜α (0) =
dt

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2. METHODS AND ANALYSIS FOR ROTATING BEC

with
F˜α (t) :=

Rd

2(|∂α ψ|2 − |∇ψ|2 ) − β|ψ|4 + (2γα2 α2 − 4Vhos )|ψ|2 + 4Ω Lz˜ (t)

+(∂y α − ∂x α) 4iΩψ ∗ (˜
x∂y + y˜∂x )ψ + 2Ω2 (˜
x2 − y˜2 )|ψ|2

d˜
x.

IV). Center of mass.
The center of mass is defined as follows:

˜ (t) =
x
Rd

˜ |ψ|2 d˜
x
x =: (˜
xc (t), y˜c (t), z˜c (t))T .

(2.10)

By [8], the center of mass satisfies a 2nd order ODE and can be solved analytically.

2.2

GPE under a rotating Lagrangian coordinate

Since the rotational term Lz is the key ‘bottle-neck’ when one derives a numerical method, we now take an orthogonal rotational transformation for (1.1) in
spatial space to deduce this rotational term and waive the difficulty. Denote the
orthogonal rotational matrix as follows:




cos(Ωt) − sin(Ωt)
A(t) := 
,
sin(Ωt) cos(Ωt)

(2.11)

if d = 2, and


if d = 3.



cos(Ωt) − sin(Ωt) 0



A(t) := 
 sin(Ωt) cos(Ωt) 0 ,



0
0
1

(2.12)

Take transformation x = A(t)˜
x, φ(x, t) = ψ(A(t)˜
x, t), and we substitute them
into (1.1). We notice that

Lz˜ψ = Lz φ, i∂t ψ = i∂t φ + ΩLz φ,

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we can therefore cancel the rotational term in (1.1) and instead solve the following problem:



 i∂t φ = − 1 ∇2 φ + V (x, t)φ + β|φ|2 φ,
2

 φ(x, 0) = φ0 (x),

where

x ∈ Rd , d = 2, 3,

V (x, t) := Vrot (x.t) + Vhos (x),

(2.13)

(2.14)

with
(γx2 − γy2 ) sin2 (Ωt)(y 2 − x2 ) + sin(2Ωt)xy
,
2

2 2
2 2

 γx x +γy y
d = 2,
2

Vrot (x.t) :=
Vhos (x) :=

γx2 x2 +γy2 y 2 +γz2 z 2
2




2.3

d = 3.

Dynamical laws in the Lagrangian coordinate

In this section, we provide some analytical results on the definition and the
dynamical laws of the following quantities for the inhomogeneous GPE (2.13):
energy, density, angular momentum expectation, condensate width and the center of mass.
I). Energy and density.
We introduce two important invariants of (2.13), which are the normalization of
the wave function:
|φ(x, t)|2 dx,

t ≥ 0,

(2.15)

E(φ) := E1 (φ) + Erot (φ),

t ≥ 0,

(2.16)

N (φ) =
Rd

and the energy

where

E1 (φ) :=

Rd

1
β
|∇φ(x, t)|2 + V (x, t)|φ(x, t)|2 + |φ(x, t)|4 dx, (2.17)
2
2
t

Erot (φ) := −

Rd

∂s V (x, s)|φ(x, s)|2 ds dx.

0

11

(2.18)


2. METHODS AND ANALYSIS FOR ROTATING BEC

We have the following theorem for the conservation of energy and density:

Theorem 2.3.1. Conservation law for the energy and density:
dE(φ)
dN (φ)
=
= 0.

dt
dt
Proof. Following the properties of the GPE under a rotating Lagrangian coordinate, we have:

N (φ) =

|φ(x)|2 dx =

|ψ(˜
x)|2 det(A) d˜
x = N (ψ).

(2.19)

Conservation of N (φ) follows directly from the conservation of N (ψ). We begin
with the equation (2.13) to show the energy conservation. Starting from
1
i∂t φ = − ∇2 φ + V (x, t)φ + β|φ|2 φ,
2

(2.20)



 i∂t φ∂t φ∗ = − 21 ∇2 φ + V (x, t)φ + β|φ|2 φ ∂t φ∗ ,

(2.21)

we have:


 −i∂ φ∗ ∂ φ = − 1 ∇2 φ∗ + V (x, t)φ∗ + β|φ|2 φ∗ ∂ φ.
t
t
t
2

Sum the two equations in (2.21) together, we obtain

0 = −
= ∂t
= ∂t
=

1
2

(∇2 φ∂t φ∗ + ∇2 φ∗ ∂t φ) dx +

Rd

Rd

1
V (x, t)∂t |φ|2 dx + ∂t
2

1
2

Rd

Rd

β
1
|∇φ(x, t)|2 + V (x, t)|φ(x, t)|2 + |φ(x, t)|4 −
2
2

|∇φ|2 dx +

β
2

Rd

|φ|4 dx + ∂t

Rd

V (x, t)|φ|2 dx −
t

β|φ|4 dx
Rd

Rd

∂t V (x, t)|φ|2 dx

∂s V (x, s)|φ(x, s)|2 ds dx

0

dE(φ)
.
dt

Hence we have the energy conservation law as stated above.

II). Angular momentum expectation.
Angular momentum is another important quantity to study the dynamics of

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